There are so many variations of this model out there (depending on time-dependencies of different parameters), but the following is the model we are going to use in this example.
Parameters alpha and sigma are constants and theta is time-dependent variable (usually calibrated to current yield curve). Swaption surface is presented in the picture below. Within the table below, time to swaption maturity has been defined in vertical axis, while tenor for underlying swap contract has been defined in horizontal axis. Co-terminal swaptions (used in calibration process) have been specifically marked with yellow colour.
Test cases for three different calibration schemes are included in the example program. More specifically, we :
- calibrate the both parameters
- calibrate sigma parameter and freeze reversion parameter to 0.05
- calibrate reversion parameter and freeze sigma parameter to 0.01
As preparatory task, builder class for constructing yield curve should be implemented into a new project from here. After this task, the following header file (ModelCalibrator.h) and tester file (Tester.cpp) should be added to this new project.
First, piecewise yield curve (USD swap curve) and swaption volatilities (co-terminal swaptions) are created by using two free functions in tester. All the data has been hard-coded inside these functions. Needless to say, in any production-level program, this data feed should come from somewhere else. After required market data and ModelCalibrator object has been created, calibration helpers for diagonal swaptions are going to be created and added to ModelCalibrator. Finally, test cases for different calibration schemes are processed.
I have noticed, that there are some rule-of-thumbs, but the actual calibration for any of these models is not so straightforward as one may think. There are some subtle issues on market data quality and products under pricing, which are leaving us with relatively high degree of freedom. Further step towards the calibration red pill can be taken by checking out this excellent research paper, written by the guys from Mizuho Securities.
Finally, as always, thanks a lot again for reading this blog.